The asymptotics of the heat semigroup for a general bounded domain with mixed boundary conditions

Small-time asymptotics of the trace of the heat semigroup theta (t) = Sigma (infinity)(nu =1), exp(-t mu (nu)); where \{mu (nu)\} are the eigenvalues of the negative Laplacian -Delta = - Sigma (2)(beta =1)(partial derivativex beta/partial derivative)(2) in the (x(1), x(2))-plane, is studied for a general bounded domain Omega with a smooth boundary partial derivative Omega, where a finite number of Dirichlet, Neumann and Robin boundary conditions, on the piecewise smooth parts Gamma (i) (i = i,..., n) of partial derivative Omega such that partial derivative Omega = U-i=1(n) Gamma (i), are considered. Some geometrical properties associated with Omega are determined.